Chapter 5.2, Problem 71E

Graphical Reasoning Consider the region bounded by the graphs of $f\left(x\right)=8x/\left(x+1\right),x=0,x=4$, and $y=0$, as shown in the figure. To print an enlarged copy of the graph, go to MathGraphs.com.

(a) Redraw the figure, and complete and shade the rectangles representing the lower sum when $n=4$. Find this lower sum.

(b) Redraw the figure, and complete and shade the rectangles representing the upper sum when $n=4$. Find this upper sum.

(c) Redraw the figure, and complete and shade the rectangles whose heights arc determined by the function values at the midpoint of each subinterval when $n=4$. Find this sum using the Midpoint Rule.

(d) Verify the following formulas for approximating the area of the region using n subintervals of equal width

Lower sum: $s\left(n\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}f\left[\left(i-1\right)\frac{4}{n}\right]}\left(\frac{4}{n}\right)$

Upper sum: $S\left(n\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}f\left[\left(i\right)\frac{4}{n}\right]}\left(\frac{4}{n}\right)$

Midpoinr rule: $M\left(n\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}f\left[\left(i-\frac{1}{2}\right)\frac{4}{n}\right]}\left(\frac{4}{n}\right)$

(e) Use a graphing utility to create a table of value of s(n), S(n) and M(n) for $n=4$, 8, 20, 100 and 200

(f) Explain why s(n) increase and S(n) decrease for increase value of n as shown in the table in part (e).